Subgroups — C₁₃×#34637 #15720976

Order 3328.

2390 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2642	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2641	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2640	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2639	C₂	1	8	yes	maximal, complemented
7	C₁₃×#2398	C₂	1	8	yes	maximal, complemented
8	C₁₃×#2297	C₂	1	4	yes	maximal, complemented
9	C₂×C₁₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×#41	C₂²	1	2	yes	—
11	C₁₃×#533	—	2	8	no	—
12	C₁₃×#532	—	2	8	no	—
13	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#125	C₂²	1	2	yes	—
17	C₂×C₁₃×#124	C₂²	1	2	yes	—
18	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
19	C₂×C₁₃×#121	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₁₃	C₂²	1	4	yes	—
22	C₂×C₁₃×#117	C₂²	1	2	yes	—
23	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
24	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×#400	C₂²	1	16	yes	—
26	C₁₃×#397	C₂²	1	16	yes	—
27	C₁₃×#394	C₂²	1	16	yes	—
28	C₁₃×#393	C₂²	1	16	yes	—
29	C₁₃×#391	C₂²	1	8	yes	—
30	C₁₃×#390	C₂²	1	8	yes	—
31	C₁₃×#386	C₂²	1	16	yes	—
32	C₄×C₁₃×C₂²⋊C₄	C₂²	1	16	yes	—
33	C₁₃×#376	C₂²	1	2	yes	—
34	C₁₃×#376	C₂²	1	2	yes	—
35	C₁₃×#376	C₂²	1	4	yes	—
36	C₂×C₁₃×#95	C₂²	1	4	yes	—
37	C₂×C₁₃×#41	—	2	8	no	—
38	C₂×C₁₃×#41	—	2	16	no	—
39	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
41	C₂²×D₄×C₁₃	C₂³	1	4	yes	—
42	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×D₄×C₁₃	C₂³	1	4	yes	—
44	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
45	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
46	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
47	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊Q₈	—	2	4	no	—
49	C₁₃×C₄⋊Q₈	—	2	8	no	—
50	C₁₃×#127	—	2	4	no	—
51	C₁₃×#125	—	2	8	no	—
52	C₁₃×#124	—	2	8	no	—
53	C₁₃×C₂²⋊Q₈	—	2	16	no	—
54	C₁₃×#121	—	2	16	no	—
55	C₄×Q₈×C₁₃	—	2	16	no	—
56	C₄×D₄×C₁₃	—	2	16	no	—
57	C₁₃×#117	—	2	8	no	—
58	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
59	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
60	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
61	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
62	C₂×C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
63	C₂×C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
64	C₂×C₁₃×C₂²⋊C₄	C₂³	1	32	yes	—
65	C₂×C₄²×C₁₃	—	2	8	no	—
66	C₂×C₄²×C₁₃	D₄	1	8	yes	—
67	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
68	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
69	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
70	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
73	C₁₃×#95	C₂³	1	32	yes	—
74	C₁₃×#95	C₂³	1	8	yes	—
75	C₁₃×#95	C₂³	1	8	yes	—
76	#34637	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂⁴×C₁₃	C₂⁴	1	4	yes	—
78	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×#41	—	4	32	no	—
80	C₂×Q₈×C₁₃	—	2	8	no	—
81	C₂×Q₈×C₁₃	—	2	4	no	—
82	C₂×Q₈×C₁₃	—	2	8	no	—
83	C₂×Q₈×C₁₃	—	4	8	no	—
84	C₂×D₄×C₁₃	—	2	8	no	—
85	C₂×D₄×C₁₃	—	2	16	no	—
86	C₂×D₄×C₁₃	—	2	8	no	—
87	C₂×D₄×C₁₃	—	2	4	no	—
88	C₂×D₄×C₁₃	—	2	8	no	—
89	C₂×D₄×C₁₃	—	4	8	no	—
90	C₂²×C₄×C₁₃	—	2	8	no	—
91	C₂²×C₄×C₁₃	—	2	8	no	—
92	C₂²×C₄×C₁₃	—	2	16	no	—
93	C₂²×C₄×C₁₃	—	4	4	no	—
94	C₂²×C₄×C₁₃	—	4	8	no	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
96	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
97	C₂²×C₄×C₁₃	#41	1	8	yes	—
98	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
99	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
100	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
101	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
102	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
103	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₁₃×C₄⋊C₄	—	2	8	no	—
106	C₁₃×C₄⋊C₄	—	2	8	no	—
107	C₁₃×C₄⋊C₄	—	2	16	no	—
108	C₁₃×C₄⋊C₄	—	2	8	no	—
109	C₁₃×C₂²⋊C₄	—	2	8	no	—
110	C₁₃×C₂²⋊C₄	—	2	16	no	—
111	C₁₃×C₂²⋊C₄	—	4	32	no	—
112	C₄²×C₁₃	—	2	8	no	—
113	C₄²×C₁₃	—	2	8	no	—
114	C₄²×C₁₃	—	2	8	no	—
115	C₄²×C₁₃	C₂×D₄	1	8	yes	—
116	C₄²×C₁₃	C₂×D₄	1	4	yes	—
117	C₄²×C₁₃	C₂×D₄	1	4	yes	—
118	C₂×#533	C₂×C₁₃	1	1	yes	characteristic, complemented
119	C₂×#532	C₂×C₁₃	1	1	yes	characteristic
120	#2642	C₂×C₁₃	1	2	yes	complemented
121	#2641	C₂×C₁₃	1	2	yes	complemented
122	#2640	C₂×C₁₃	1	4	yes	complemented
123	#2639	C₂×C₁₃	1	8	yes	complemented
124	#2398	C₂×C₁₃	1	8	yes	complemented
125	#2297	C₂×C₁₃	1	4	yes	complemented
126	C₂×#376	C₂×C₁₃	1	1	yes	characteristic, complemented
127	C₂³×C₁₃	—	2	4	no	—
128	C₂³×C₁₃	—	2	4	no	—
129	C₂³×C₁₃	—	2	8	no	—
130	C₂³×C₁₃	—	2	16	no	—
131	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
132	C₂³×C₁₃	C₂×#41	1	4	yes	—
133	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
134	Q₈×C₁₃	—	4	16	no	—
135	Q₈×C₁₃	—	4	8	no	—
136	D₄×C₁₃	—	4	16	no	—
137	D₄×C₁₃	—	4	16	no	—
138	D₄×C₁₃	—	4	8	no	—
139	C₂×C₄×C₁₃	—	2	4	no	—
140	C₂×C₄×C₁₃	—	2	4	no	—
141	C₂×C₄×C₁₃	—	2	8	no	—
142	C₂×C₄×C₁₃	—	2	4	no	—
143	C₂×C₄×C₁₃	—	2	8	no	—
144	C₂×C₄×C₁₃	—	2	8	no	—
145	C₂×C₄×C₁₃	—	2	16	no	—
146	C₂×C₄×C₁₃	—	2	16	no	—
147	C₂×C₄×C₁₃	—	4	8	no	—
148	C₂×C₄×C₁₃	—	4	16	no	—
149	C₂×C₄×C₁₃	—	4	16	no	—
150	C₂×C₄×C₁₃	#124	1	16	yes	—
151	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
152	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
153	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
154	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
155	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
156	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
157	C₂²×#41	C₂²×C₁₃	1	2	yes	—
158	#533	—	2	8	no	—
159	#532	—	2	8	no	—
160	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
161	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×#125	C₂²×C₁₃	1	2	yes	—
164	C₂×#124	C₂²×C₁₃	1	2	yes	—
165	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
166	C₂×#121	C₂²×C₁₃	1	4	yes	—
167	C₂×C₄×Q₈	C₂²×C₁₃	1	4	yes	—
168	C₂×C₄×D₄	C₂²×C₁₃	1	4	yes	—
169	C₂×#117	C₂²×C₁₃	1	2	yes	—
170	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
171	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
172	#400	C₂²×C₁₃	1	16	yes	—
173	#397	C₂²×C₁₃	1	16	yes	—
174	#394	C₂²×C₁₃	1	16	yes	—
175	#393	C₂²×C₁₃	1	16	yes	—
176	#391	C₂²×C₁₃	1	8	yes	—
177	#390	C₂²×C₁₃	1	8	yes	—
178	#386	C₂²×C₁₃	1	16	yes	—
179	C₄×C₂²⋊C₄	C₂²×C₁₃	1	16	yes	—
180	#376	C₂²×C₁₃	1	2	yes	—
181	#376	C₂²×C₁₃	1	2	yes	—
182	#376	C₂²×C₁₃	1	4	yes	—
183	C₂×#95	C₂²×C₁₃	1	4	yes	—
184	C₂²×C₁₃	—	2	4	no	—
185	C₂²×C₁₃	—	2	8	no	—
186	C₂²×C₁₃	—	4	4	no	—
187	C₂²×C₁₃	—	4	8	no	—
188	C₂²×C₁₃	—	4	16	no	—
189	C₂²×C₁₃	C₂×#124	1	4	yes	—
190	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
191	C₂²×C₁₃	C₂²×#41	1	2	yes	—
192	C₂²×C₁₃	#585	1	4	yes	—
193	C₄×C₁₃	—	2	4	no	—
194	C₄×C₁₃	—	2	4	no	—
195	C₄×C₁₃	—	4	4	no	—
196	C₄×C₁₃	—	4	8	no	—
197	C₄×C₁₃	—	4	8	no	—
198	C₄×C₁₃	C₂×#124	1	4	yes	—
199	C₄×C₁₃	C₂×#124	1	4	yes	—
200	C₂×#41	—	2	8	no	—
201	C₂×#41	—	2	16	no	—
202	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
203	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
204	C₂²×D₄	C₂³×C₁₃	1	4	yes	—
205	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
206	C₂²×D₄	C₂³×C₁₃	1	4	yes	—
207	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
208	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
209	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
210	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	4	no	—
212	C₄⋊Q₈	—	2	8	no	—
213	#127	—	2	4	no	—
214	#125	—	2	8	no	—
215	#124	—	2	8	no	—
216	C₂²⋊Q₈	—	2	16	no	—
217	#121	—	2	16	no	—
218	C₄×Q₈	—	2	16	no	—
219	C₄×D₄	—	2	16	no	—
220	#117	—	2	8	no	—
221	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
222	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
223	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
224	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
225	C₂×C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
226	C₂×C₂²⋊C₄	C₂³×C₁₃	1	8	yes	—
227	C₂×C₂²⋊C₄	C₂³×C₁₃	1	32	yes	—
228	C₂×C₄²	—	2	8	no	—
229	C₂×C₄²	D₄×C₁₃	1	8	yes	—
230	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
231	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
232	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
233	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
234	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
236	#95	C₂³×C₁₃	1	8	yes	—
237	#95	C₂³×C₁₃	1	8	yes	—
238	#95	C₂³×C₁₃	1	32	yes	—
239	C₂×C₁₃	—	2	4	no	—
240	C₂×C₁₃	—	8	4	no	—
241	C₂×C₁₃	C₂²×#124	1	1	yes	characteristic
242	C₂×C₁₃	#3449	1	2	yes	—
243	C₂×C₁₃	#3455	1	4	yes	—
244	C₂⁴	C₂⁴×C₁₃	1	4	yes	—
245	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
246	#41	—	4	32	no	—
247	C₂×Q₈	—	2	8	no	—
248	C₂×Q₈	—	2	4	no	—
249	C₂×Q₈	—	2	8	no	—
250	C₂×Q₈	—	4	8	no	—
251	C₂×D₄	—	2	8	no	—
252	C₂×D₄	—	2	16	no	—
253	C₂×D₄	—	2	8	no	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	2	8	no	—
256	C₂×D₄	—	4	8	no	—
257	C₂²×C₄	—	2	8	no	—
258	C₂²×C₄	—	2	8	no	—
259	C₂²×C₄	—	2	16	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	—	4	8	no	—
262	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
263	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
264	C₂²×C₄	C₁₃×#41	1	8	yes	—
265	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
266	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
267	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
268	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
269	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
270	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
271	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	8	no	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	16	no	—
275	C₄⋊C₄	—	2	8	no	—
276	C₂²⋊C₄	—	2	8	no	—
277	C₂²⋊C₄	—	2	16	no	—
278	C₂²⋊C₄	—	4	32	no	—
279	C₄²	—	2	8	no	—
280	C₄²	—	2	8	no	—
281	C₄²	—	2	8	no	—
282	C₄²	C₂×D₄×C₁₃	1	8	yes	—
283	C₄²	C₂×D₄×C₁₃	1	4	yes	—
284	C₄²	C₂×D₄×C₁₃	1	4	yes	—
285	C₁₃	#34637	1	1	yes	characteristic, direct product factor, Sylow
286	C₂³	—	2	4	no	—
287	C₂³	—	2	4	no	—
288	C₂³	—	2	8	no	—
289	C₂³	—	2	16	no	—
290	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
291	C₂³	C₂×C₁₃×#41	1	4	yes	—
292	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
293	Q₈	—	4	16	no	—
294	Q₈	—	4	8	no	—
295	D₄	—	4	16	no	—
296	D₄	—	4	16	no	—
297	D₄	—	4	8	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	2	16	no	—
305	C₂×C₄	—	2	16	no	—
306	C₂×C₄	—	4	8	no	—
307	C₂×C₄	—	4	16	no	—
308	C₂×C₄	—	4	16	no	—
309	C₂×C₄	C₁₃×#124	1	16	yes	—
310	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
311	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
312	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
313	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
314	C₂×C₄	C₂×C₁₃×#41	1	4	yes	—
315	C₂×C₄	C₂×C₁₃×#41	1	4	yes	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	8	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	4	16	no	—
321	C₂²	C₂×C₁₃×#124	1	4	yes	—
322	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
323	C₂²	C₂²×C₁₃×#41	1	2	yes	—
324	C₂²	C₁₃×#585	1	4	yes	—
325	C₄	—	2	4	no	—
326	C₄	—	2	4	no	—
327	C₄	—	4	4	no	—
328	C₄	—	4	8	no	—
329	C₄	—	4	8	no	—
330	C₄	C₂×C₁₃×#124	1	4	yes	—
331	C₄	C₂×C₁₃×#124	1	4	yes	—
332	C₂	—	2	4	no	—
333	C₂	—	8	4	no	—
334	C₂	C₂²×C₁₃×#124	1	1	yes	characteristic
335	C₂	C₁₃×#3449	1	2	yes	—
336	C₂	C₁₃×#3455	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
socle	C₂³×C₁₃	C₂⁵
prime radical	C₂×C₁₃×#532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₁₃×#533	C₂	semidirect
3. C₁₃×#2642	C₂	semidirect
4. C₁₃×#2641	C₂	semidirect
5. C₁₃×#2640	C₂	semidirect
6. C₁₃×#2639	C₂	semidirect
7. C₁₃×#2398	C₂	semidirect
8. C₁₃×#2297	C₂	semidirect
9. C₂×C₁₃×#376	C₂	semidirect
76. #34637	C₁₃	direct (N × H)
118. C₂×#533	C₂×C₁₃	semidirect
120. #2642	C₂×C₁₃	semidirect
121. #2641	C₂×C₁₃	semidirect
122. #2640	C₂×C₁₃	semidirect
123. #2639	C₂×C₁₃	semidirect
124. #2398	C₂×C₁₃	semidirect
125. #2297	C₂×C₁₃	semidirect
126. C₂×#376	C₂×C₁₃	semidirect
285. C₁₃	#34637	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#533	2
C₂×C₁₃×#532	2
C₁₃×#2642	2
C₁₃×#2641	2
C₁₃×#2640	2
C₁₃×#2639	2
C₁₃×#2398	2
C₁₃×#2297	2
C₂×C₁₃×#376	2
#34637	13

Subgroups — C13×#34637 #15720976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#34637 #15720976